answer：To find the general form of the line's equation from the given parametric equations:1. **Express the parametric equations:** [ begin{cases} x = 2 - 3t y = 1 + 2t end{cases} ]2. **Solve for the parameter ( t ) in both equations:** - From the first equation: [ x = 2 - 3t implies 3t = 2 - x implies t = frac{2 - x}{3} ] - From the second equation: [ y = 1 + 2t implies 2t = y - 1 implies t = frac{y - 1}{2} ]3. **Set the expressions for ( t ) equal to each other:** [ frac{2 - x}{3} = frac{y - 1}{2} ]4. **Cross-multiply to eliminate the denominators:** [ 2(2 - x) = 3(y - 1) ]5. **Expand and simplify the equation:** [ 4 - 2x = 3y - 3 ] [ -2x - 3y + 7 = 0 ] [ 2x + 3y - 7 = 0 ]6. **Final Answer:** [ boxed{2x + 3y - 7 = 0} ]

answer：First, identify the distinct marbles: red, green, blue, orange, and five identical yellow marbles.To find the number of different groups of two marbles, consider two cases:1. **Both marbles are yellow**: Since all yellow marbles are identical, there is only 1 way to choose two yellow marbles.2. **One marble is yellow and the other is a different color**: There are 4 distinct colors (red, green, blue, orange) to pair with a yellow marble. This gives 4 different groups.3. **Both marbles are of different colors (non-yellow)**: Calculate the number of ways to choose 2 distinct colors from the 4 available. This is done using the combination formula C(4,2), which equals 6.Finally, add up all the possible groups: 1 (both yellow) + 4 (one yellow and one colored) + 6 (both colored) = 11 different groups.

answer：Okay, so I have this expression here: (3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}}), and it says that the 3s and 5s alternate in the numerator, and the 1s and 2s alternate in the denominator. I need to find what real number this expression equals. Hmm, it looks like a continued fraction, right? Let me try to understand the pattern here. The expression starts with 3, then in the numerator, it alternates between 3 and 5, and in the denominator, it alternates between 1 and 2. So, after the first 3, the next term is 3 over (1 plus 5 over (2 plus 3 over (1 plus ...))). It seems like it's repeating a pattern of 3, 5, 3, 5 in the numerators and 1, 2, 1, 2 in the denominators.I remember that for continued fractions, sometimes we can set the entire expression equal to a variable and then solve for that variable. Maybe I can do something similar here. Let me denote the entire expression as (x). So, (x = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}}). Wait, but looking at the structure, after the first 3, the rest of the fraction seems to repeat the same pattern. So, if I look at the part after the first 3, which is (frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}}), maybe that part can also be expressed in terms of (x). Let me see.If I let (x = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}}), then the part after the first 3 is (frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}}). But the denominator here is (1 + frac{5}{2 + frac{3}{1 + cdots}}). Hmm, is that similar to the original expression?Wait, the original expression starts with 3, then has a 3 in the numerator, then 1 in the denominator, then 5 in the numerator, then 2 in the denominator, and so on. So, if I look at the part after the first 3, it's (frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}}). Let me denote this part as (y), so (y = frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}}). Then, (x = 3 + y).Now, looking at (y), it's (frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}}). The denominator here is (1 + frac{5}{2 + frac{3}{1 + cdots}}). Let me denote the part after the 1 as (z), so (z = frac{5}{2 + frac{3}{1 + cdots}}). Then, the denominator becomes (1 + z), so (y = frac{3}{1 + z}).Now, looking at (z), it's (frac{5}{2 + frac{3}{1 + cdots}}). The denominator here is (2 + frac{3}{1 + cdots}). Let me denote the part after the 2 as (w), so (w = frac{3}{1 + cdots}). Then, the denominator becomes (2 + w), so (z = frac{5}{2 + w}).Wait a minute, but (w = frac{3}{1 + cdots}), which looks similar to the original expression (x). Because (x = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}}), so the part after the first 3 is (frac{3}{1 + frac{5}{2 + cdots}}), which is similar to (w). So, maybe (w) is related to (x) somehow.Let me try to express all these variables in terms of each other. So, starting from the beginning:1. (x = 3 + y)2. (y = frac{3}{1 + z})3. (z = frac{5}{2 + w})4. (w = frac{3}{1 + cdots})But the (cdots) in (w) is the same as the original expression after the first 3, which is (y). So, (w = y). Therefore, (w = y).So, substituting back:4. (w = y)3. (z = frac{5}{2 + w} = frac{5}{2 + y})2. (y = frac{3}{1 + z} = frac{3}{1 + frac{5}{2 + y}})1. (x = 3 + y)So, now we have an equation for (y) in terms of (y). Let me write that out:(y = frac{3}{1 + frac{5}{2 + y}})Let me solve this equation for (y). First, simplify the denominator:(1 + frac{5}{2 + y} = frac{(2 + y) + 5}{2 + y} = frac{7 + y}{2 + y})So, (y = frac{3}{frac{7 + y}{2 + y}} = 3 cdot frac{2 + y}{7 + y})So, (y = frac{3(2 + y)}{7 + y})Multiply both sides by (7 + y):(y(7 + y) = 3(2 + y))Expand both sides:(7y + y^2 = 6 + 3y)Bring all terms to one side:(y^2 + 7y - 3y - 6 = 0)Simplify:(y^2 + 4y - 6 = 0)Now, solve this quadratic equation for (y). Using the quadratic formula:(y = frac{-4 pm sqrt{16 + 24}}{2} = frac{-4 pm sqrt{40}}{2} = frac{-4 pm 2sqrt{10}}{2} = -2 pm sqrt{10})Since (y) is a positive number (as it's part of a continued fraction with positive terms), we take the positive root:(y = -2 + sqrt{10})Wait, but (-2 + sqrt{10}) is approximately (-2 + 3.16 = 1.16), which is positive, so that's okay.Now, recall that (x = 3 + y), so:(x = 3 + (-2 + sqrt{10}) = 1 + sqrt{10})Wait, but let me double-check my steps because I might have made a mistake. Let me go back to the equation for (y):(y = frac{3}{1 + frac{5}{2 + y}})I simplified the denominator to (frac{7 + y}{2 + y}), which seems correct. Then, (y = 3 cdot frac{2 + y}{7 + y}), which is also correct.Multiplying both sides by (7 + y):(y(7 + y) = 3(2 + y))Expanding:(7y + y^2 = 6 + 3y)Bringing all terms to one side:(y^2 + 4y - 6 = 0)Yes, that's correct. Then, applying the quadratic formula:(y = frac{-4 pm sqrt{16 + 24}}{2} = frac{-4 pm sqrt{40}}{2} = frac{-4 pm 2sqrt{10}}{2} = -2 pm sqrt{10})So, (y = -2 + sqrt{10}) since the other root is negative.Therefore, (x = 3 + y = 3 + (-2 + sqrt{10}) = 1 + sqrt{10})Wait, but let me check if this makes sense. If I plug (x = 1 + sqrt{10}) back into the original expression, does it hold?Let me compute (x = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}})If (x = 1 + sqrt{10}), then the continued fraction part should equal (x - 3 = -2 + sqrt{10}), which is approximately 1.16, as before.Let me compute the continued fraction part:(frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}})If I denote this as (y), then (y = frac{3}{1 + frac{5}{2 + y}}), which we already solved and got (y = -2 + sqrt{10}). So, yes, it seems consistent.Wait, but let me compute (1 + sqrt{10}) numerically to see if it makes sense. (sqrt{10}) is approximately 3.1623, so (1 + sqrt{10}) is approximately 4.1623.Let me compute the continued fraction numerically to see if it converges to that.Start with an initial guess for the continued fraction. Let's say we start with the innermost part as 1, then compute upwards.First iteration:Start with 1.Compute (2 + frac{3}{1} = 2 + 3 = 5)Then, compute (1 + frac{5}{5} = 1 + 1 = 2)Then, compute (frac{3}{2} = 1.5)Then, add 3: (3 + 1.5 = 4.5)Second iteration:Start with 1.Compute (2 + frac{3}{1} = 5)Compute (1 + frac{5}{5} = 2)Compute (frac{3}{2} = 1.5)Add 3: 4.5Wait, that's the same as before. Maybe I need to go deeper.Alternatively, let's compute more iterations.Let me define the continued fraction as follows:Let me denote (a_1 = 3), (a_2 = 5), (a_3 = 3), (a_4 = 5), etc., alternating between 3 and 5.Similarly, the denominators alternate between 1 and 2: (b_1 = 1), (b_2 = 2), (b_3 = 1), (b_4 = 2), etc.So, the continued fraction can be written as:(x = a_1 + frac{a_2}{b_1 + frac{a_3}{b_2 + frac{a_4}{b_3 + cdots}}})Which is:(x = 3 + frac{5}{1 + frac{3}{2 + frac{5}{1 + frac{3}{2 + cdots}}}})Wait, but in the original problem, the numerators alternate between 3 and 5, starting with 3, then 5, then 3, etc., and the denominators alternate between 1 and 2, starting with 1, then 2, then 1, etc.So, the continued fraction is:(x = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + cdots}}}})Wait, so the numerators are 3, 5, 3, 5,... and denominators are 1, 2, 1, 2,...So, in the first step, it's 3 over (1 + 5 over (2 + 3 over (1 + 5 over (2 + ...))))So, perhaps I need to adjust my earlier approach.Let me try setting (x = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}})Let me denote the continued fraction part as (y), so (y = frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}})Then, (x = 3 + y)Now, looking at (y), it's (frac{3}{1 + frac{5}{2 + frac{3}{1 + cdots}}}). Let me denote the denominator as (z = 1 + frac{5}{2 + frac{3}{1 + cdots}}), so (y = frac{3}{z})Now, (z = 1 + frac{5}{2 + frac{3}{1 + cdots}}). Let me denote the denominator here as (w = 2 + frac{3}{1 + cdots}), so (z = 1 + frac{5}{w})Now, (w = 2 + frac{3}{1 + cdots}). The (cdots) here is the same as the original continued fraction part (y), because after the 1, it's the same pattern: 3 over (1 + 5 over (2 + ...)). So, (w = 2 + frac{3}{1 + cdots} = 2 + frac{3}{1 + frac{5}{2 + cdots}}). Wait, but that's similar to (z), which is (1 + frac{5}{w}).Wait, this is getting a bit tangled. Let me try to express everything in terms of (y).We have:1. (x = 3 + y)2. (y = frac{3}{z})3. (z = 1 + frac{5}{w})4. (w = 2 + frac{3}{v})5. (v = 1 + frac{5}{u})6. (u = 2 + frac{3}{t})... and so on.But this seems like it's going on infinitely, so perhaps I need a different approach.Wait, maybe I can express (w) in terms of (z). Let me see.From step 3: (z = 1 + frac{5}{w})From step 4: (w = 2 + frac{3}{v})But (v) is similar to (z), because (v = 1 + frac{5}{u}), and (u = 2 + frac{3}{t}), etc. So, perhaps (v = z), because the pattern repeats.Wait, let me think. If I look at the structure, after the first few terms, the pattern repeats. So, perhaps (v = z), (u = w), etc.So, assuming that (v = z), then from step 4: (w = 2 + frac{3}{v} = 2 + frac{3}{z})And from step 3: (z = 1 + frac{5}{w} = 1 + frac{5}{2 + frac{3}{z}})So, now we have an equation involving (z):(z = 1 + frac{5}{2 + frac{3}{z}})Let me solve this equation for (z).First, simplify the denominator:(2 + frac{3}{z} = frac{2z + 3}{z})So, (z = 1 + frac{5}{frac{2z + 3}{z}} = 1 + frac{5z}{2z + 3})Now, multiply both sides by (2z + 3) to eliminate the denominator:(z(2z + 3) = (1)(2z + 3) + 5z)Expand the left side:(2z^2 + 3z = 2z + 3 + 5z)Simplify the right side:(2z^2 + 3z = 7z + 3)Bring all terms to one side:(2z^2 + 3z - 7z - 3 = 0)Simplify:(2z^2 - 4z - 3 = 0)Now, solve this quadratic equation for (z). Using the quadratic formula:(z = frac{4 pm sqrt{16 + 24}}{4} = frac{4 pm sqrt{40}}{4} = frac{4 pm 2sqrt{10}}{4} = frac{2 pm sqrt{10}}{2})Since (z) must be positive, we take the positive root:(z = frac{2 + sqrt{10}}{2} = 1 + frac{sqrt{10}}{2})Now, recall that (y = frac{3}{z} = frac{3}{1 + frac{sqrt{10}}{2}} = frac{3}{frac{2 + sqrt{10}}{2}} = frac{6}{2 + sqrt{10}})To rationalize the denominator, multiply numerator and denominator by (2 - sqrt{10}):(y = frac{6(2 - sqrt{10})}{(2 + sqrt{10})(2 - sqrt{10})} = frac{12 - 6sqrt{10}}{4 - 10} = frac{12 - 6sqrt{10}}{-6} = -2 + sqrt{10})So, (y = -2 + sqrt{10}), which is approximately 1.1623.Therefore, (x = 3 + y = 3 + (-2 + sqrt{10}) = 1 + sqrt{10})Wait, but earlier I thought (x = 1 + sqrt{10}), but when I tried plugging in the continued fraction numerically, I got 4.5 after one iteration, which is higher than 1 + sqrt(10) ≈ 4.1623. So, maybe it's converging towards 4.1623.Let me check with more iterations.Let me compute the continued fraction step by step:Start with an initial guess for the innermost part. Let's say we start with 1.First iteration:Compute (2 + frac{3}{1} = 5)Then, compute (1 + frac{5}{5} = 2)Then, compute (frac{3}{2} = 1.5)Add 3: (3 + 1.5 = 4.5)Second iteration:Start with 1.Compute (2 + frac{3}{1} = 5)Compute (1 + frac{5}{5} = 2)Compute (frac{3}{2} = 1.5)Add 3: 4.5Wait, same result. Maybe I need to go deeper.Alternatively, let's compute more iterations by considering more layers.Let me define the continued fraction as follows:Let me denote the continued fraction as (C = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + cdots}}}})Let me compute (C) by approximating it with a few layers.First approximation: (C_1 = 3 + frac{3}{1} = 6)Second approximation: (C_2 = 3 + frac{3}{1 + frac{5}{2}} = 3 + frac{3}{1 + 2.5} = 3 + frac{3}{3.5} ≈ 3 + 0.8571 ≈ 3.8571)Third approximation: (C_3 = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1}}} = 3 + frac{3}{1 + frac{5}{5}} = 3 + frac{3}{1 + 1} = 3 + 1.5 = 4.5)Fourth approximation: (C_4 = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2}}}})Compute the innermost part: (1 + frac{5}{2} = 3.5)Then, compute (2 + frac{3}{3.5} ≈ 2 + 0.8571 ≈ 2.8571)Then, compute (1 + frac{5}{2.8571} ≈ 1 + 1.75 ≈ 2.75)Then, compute (frac{3}{2.75} ≈ 1.0909)Add 3: (3 + 1.0909 ≈ 4.0909)Fifth approximation: (C_5 = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1}}}}})Compute the innermost part: (1 + frac{5}{2 + 3} = 1 + 1 = 2)Then, compute (2 + frac{3}{2} = 3.5)Then, compute (1 + frac{5}{3.5} ≈ 1 + 1.4286 ≈ 2.4286)Then, compute (frac{3}{2.4286} ≈ 1.2353)Add 3: (3 + 1.2353 ≈ 4.2353)Sixth approximation: (C_6 = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2}}}}}})Compute the innermost part: (1 + frac{5}{2} = 3.5)Then, compute (2 + frac{3}{3.5} ≈ 2.8571)Then, compute (1 + frac{5}{2.8571} ≈ 2.75)Then, compute (2 + frac{3}{2.75} ≈ 2 + 1.0909 ≈ 3.0909)Then, compute (1 + frac{5}{3.0909} ≈ 1 + 1.6176 ≈ 2.6176)Then, compute (frac{3}{2.6176} ≈ 1.146)Add 3: (3 + 1.146 ≈ 4.146)Seventh approximation: (C_7 = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1}}}}}}})Compute the innermost part: (1 + frac{5}{2 + 3} = 2)Then, compute (2 + frac{3}{2} = 3.5)Then, compute (1 + frac{5}{3.5} ≈ 2.4286)Then, compute (2 + frac{3}{2.4286} ≈ 3.2353)Then, compute (1 + frac{5}{3.2353} ≈ 1 + 1.545 ≈ 2.545)Then, compute (frac{3}{2.545} ≈ 1.179)Add 3: (3 + 1.179 ≈ 4.179)Eighth approximation: (C_8 = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2}}}}}}}})Compute the innermost part: (1 + frac{5}{2} = 3.5)Then, compute (2 + frac{3}{3.5} ≈ 2.8571)Then, compute (1 + frac{5}{2.8571} ≈ 2.75)Then, compute (2 + frac{3}{2.75} ≈ 3.0909)Then, compute (1 + frac{5}{3.0909} ≈ 2.6176)Then, compute (2 + frac{3}{2.6176} ≈ 3.146)Then, compute (1 + frac{5}{3.146} ≈ 1 + 1.59 ≈ 2.59)Then, compute (frac{3}{2.59} ≈ 1.158)Add 3: (3 + 1.158 ≈ 4.158)Ninth approximation: (C_9 = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1}}}}}}}}})Compute the innermost part: (1 + frac{5}{2 + 3} = 2)Then, compute (2 + frac{3}{2} = 3.5)Then, compute (1 + frac{5}{3.5} ≈ 2.4286)Then, compute (2 + frac{3}{2.4286} ≈ 3.2353)Then, compute (1 + frac{5}{3.2353} ≈ 2.545)Then, compute (2 + frac{3}{2.545} ≈ 3.179)Then, compute (1 + frac{5}{3.179} ≈ 2.589)Then, compute (frac{3}{2.589} ≈ 1.159)Add 3: (3 + 1.159 ≈ 4.159)Tenth approximation: (C_{10} = 3 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2 + frac{3}{1 + frac{5}{2}}}}}}}}}})Compute the innermost part: (1 + frac{5}{2} = 3.5)Then, compute (2 + frac{3}{3.5} ≈ 2.8571)Then, compute (1 + frac{5}{2.8571} ≈ 2.75)Then, compute (2 + frac{3}{2.75} ≈ 3.0909)Then, compute (1 + frac{5}{3.0909} ≈ 2.6176)Then, compute (2 + frac{3}{2.6176} ≈ 3.146)Then, compute (1 + frac{5}{3.146} ≈ 2.59)Then, compute (2 + frac{3}{2.59} ≈ 3.158)Then, compute (1 + frac{5}{3.158} ≈ 2.589)Then, compute (frac{3}{2.589} ≈ 1.159)Add 3: (3 + 1.159 ≈ 4.159)So, after ten approximations, the value seems to be converging around 4.159, which is close to (1 + sqrt{10}) since (sqrt{10} ≈ 3.1623), so (1 + sqrt{10} ≈ 4.1623). So, that seems consistent.Therefore, my initial solution that (x = 1 + sqrt{10}) seems correct.But let me double-check my algebra when I set up the equation for (z). I had:(z = 1 + frac{5}{2 + frac{3}{z}})Then, I simplified it to:(z = 1 + frac{5z}{2z + 3})Wait, let me verify that step.Starting with:(z = 1 + frac{5}{2 + frac{3}{z}})First, compute the denominator (2 + frac{3}{z} = frac{2z + 3}{z})So, (z = 1 + frac{5}{frac{2z + 3}{z}} = 1 + frac{5z}{2z + 3})Yes, that's correct.Then, multiply both sides by (2z + 3):(z(2z + 3) = (1)(2z + 3) + 5z)Which gives:(2z^2 + 3z = 2z + 3 + 5z)Simplify the right side:(2z^2 + 3z = 7z + 3)Bring all terms to one side:(2z^2 + 3z - 7z - 3 = 0)Which simplifies to:(2z^2 - 4z - 3 = 0)Yes, that's correct.Then, solving for (z):(z = frac{4 pm sqrt{16 + 24}}{4} = frac{4 pm sqrt{40}}{4} = frac{4 pm 2sqrt{10}}{4} = frac{2 pm sqrt{10}}{2})Since (z) must be positive, we take (z = frac{2 + sqrt{10}}{2} = 1 + frac{sqrt{10}}{2})Then, (y = frac{3}{z} = frac{3}{1 + frac{sqrt{10}}{2}} = frac{6}{2 + sqrt{10}})Rationalizing the denominator:(y = frac{6(2 - sqrt{10})}{(2 + sqrt{10})(2 - sqrt{10})} = frac{12 - 6sqrt{10}}{4 - 10} = frac{12 - 6sqrt{10}}{-6} = -2 + sqrt{10})So, (y = -2 + sqrt{10}), which is approximately 1.1623.Then, (x = 3 + y = 3 + (-2 + sqrt{10}) = 1 + sqrt{10}), which is approximately 4.1623.All the steps seem correct, and the numerical approximations support this result. Therefore, I can conclude that the value of the expression is (1 + sqrt{10}).

answer：Okay, so I have this problem here: "At what point does the line (5y + 3x = 15) intersect the (x)-axis? Express your answer as an ordered pair." Hmm, I need to find where this line crosses the x-axis. I remember that when a line intersects the x-axis, the y-coordinate at that point is zero. So, I think I should set (y = 0) in the equation and solve for (x).Let me write that down. If (y = 0), then the equation becomes:[5(0) + 3x = 15]Simplifying that, I get:[0 + 3x = 15]So, (3x = 15). To find (x), I need to divide both sides by 3:[x = frac{15}{3}][x = 5]Okay, so when (y = 0), (x = 5). That means the point where the line intersects the x-axis is ((5, 0)). Let me double-check to make sure I didn't make any mistakes. Plugging (x = 5) and (y = 0) back into the original equation:[5(0) + 3(5) = 15][0 + 15 = 15][15 = 15]Yep, that works out. So, I think I've got it right. The line intersects the x-axis at the point ((5, 0)).

answer：Okay, so I have this problem where I'm given two quadratic equations: ( ax^2 + bx + c = 0 ) and ( cx^2 + bx + a = 0 ). The first equation has two distinct real solutions ( p_1 ) and ( p_2 ), and the second equation has two distinct real solutions ( q_1 ) and ( q_2 ). It's also given that these four numbers ( p_1, q_1, p_2, q_2 ) form an arithmetic progression in that specific order. I need to show that ( a + c = 0 ).Alright, let me start by recalling what an arithmetic progression (AP) is. In an AP, the difference between consecutive terms is constant. So, if ( p_1, q_1, p_2, q_2 ) are in AP, then the difference between ( q_1 ) and ( p_1 ) is the same as the difference between ( p_2 ) and ( q_1 ), and so on.Let me denote the common difference by ( d ). So, I can write:( q_1 = p_1 + d )( p_2 = q_1 + d = p_1 + 2d )( q_2 = p_2 + d = p_1 + 3d )So, the four terms are ( p_1, p_1 + d, p_1 + 2d, p_1 + 3d ).Now, let's think about the properties of quadratic equations. For the first equation ( ax^2 + bx + c = 0 ), the sum of the roots ( p_1 + p_2 = -frac{b}{a} ) and the product ( p_1 p_2 = frac{c}{a} ). Similarly, for the second equation ( cx^2 + bx + a = 0 ), the sum of the roots ( q_1 + q_2 = -frac{b}{c} ) and the product ( q_1 q_2 = frac{a}{c} ).Given that ( p_1, q_1, p_2, q_2 ) are in AP, let's express their sums and products in terms of ( p_1 ) and ( d ).First, the sum of the roots for the first equation:( p_1 + p_2 = p_1 + (p_1 + 2d) = 2p_1 + 2d = -frac{b}{a} ) ...(1)Similarly, the sum of the roots for the second equation:( q_1 + q_2 = (p_1 + d) + (p_1 + 3d) = 2p_1 + 4d = -frac{b}{c} ) ...(2)So, now I have two equations:1. ( 2p_1 + 2d = -frac{b}{a} )2. ( 2p_1 + 4d = -frac{b}{c} )Let me subtract equation (1) from equation (2):( (2p_1 + 4d) - (2p_1 + 2d) = -frac{b}{c} - (-frac{b}{a}) )Simplifying the left side:( 2d = -frac{b}{c} + frac{b}{a} )Factor out ( b ):( 2d = b left( frac{1}{a} - frac{1}{c} right) )So,( 2d = b left( frac{c - a}{ac} right) )Which can be rewritten as:( 2d = frac{b(c - a)}{ac} )Now, since ( d ) is the common difference in the arithmetic progression, it can't be zero because that would mean all terms are equal, which contradicts the fact that the quadratics have distinct roots. So, ( d neq 0 ).Therefore, I can solve for ( frac{c - a}{ac} ):( frac{c - a}{ac} = frac{2d}{b} )Let me denote ( k = frac{2d}{b} ), so:( frac{c - a}{ac} = k )Which implies:( c - a = kac )Hmm, not sure if that helps directly. Maybe I need another approach.Wait, let's think about the products of the roots.For the first equation, the product is ( p_1 p_2 = frac{c}{a} ). Substituting ( p_2 = p_1 + 2d ):( p_1 (p_1 + 2d) = frac{c}{a} )Similarly, for the second equation, the product is ( q_1 q_2 = frac{a}{c} ). Substituting ( q_1 = p_1 + d ) and ( q_2 = p_1 + 3d ):( (p_1 + d)(p_1 + 3d) = frac{a}{c} )Let me expand both products.First, ( p_1 (p_1 + 2d) = p_1^2 + 2d p_1 = frac{c}{a} ) ...(3)Second, ( (p_1 + d)(p_1 + 3d) = p_1^2 + 4d p_1 + 3d^2 = frac{a}{c} ) ...(4)Now, let's subtract equation (3) from equation (4):( (p_1^2 + 4d p_1 + 3d^2) - (p_1^2 + 2d p_1) = frac{a}{c} - frac{c}{a} )Simplify left side:( 2d p_1 + 3d^2 = frac{a}{c} - frac{c}{a} )Factor the right side:( frac{a^2 - c^2}{ac} = frac{(a - c)(a + c)}{ac} )So, we have:( 2d p_1 + 3d^2 = frac{(a - c)(a + c)}{ac} )Hmm, this seems a bit complicated. Maybe I can express ( p_1 ) from equation (1) and substitute it here.From equation (1):( 2p_1 + 2d = -frac{b}{a} )So,( p_1 = -frac{b}{2a} - d )Let me substitute this into the equation above:( 2d left( -frac{b}{2a} - d right) + 3d^2 = frac{(a - c)(a + c)}{ac} )Simplify the left side:First term: ( 2d times -frac{b}{2a} = -frac{b d}{a} )Second term: ( 2d times (-d) = -2d^2 )Third term: ( +3d^2 )So, combining:( -frac{b d}{a} - 2d^2 + 3d^2 = -frac{b d}{a} + d^2 )Therefore, the equation becomes:( -frac{b d}{a} + d^2 = frac{(a - c)(a + c)}{ac} )Let me factor the right side:( frac{(a - c)(a + c)}{ac} = frac{a^2 - c^2}{ac} )So,( -frac{b d}{a} + d^2 = frac{a^2 - c^2}{ac} )Let me multiply both sides by ( ac ) to eliminate denominators:Left side:( -frac{b d}{a} times ac = -b d c )( d^2 times ac = a c d^2 )So, left side becomes:( -b c d + a c d^2 )Right side:( frac{a^2 - c^2}{ac} times ac = a^2 - c^2 )So, the equation is:( -b c d + a c d^2 = a^2 - c^2 )Let me rearrange this:( a c d^2 - b c d - a^2 + c^2 = 0 )Hmm, this is a quadratic in terms of ( d ), but it's getting complicated. Maybe I need to find another relation.Wait, earlier I had:( 2d = frac{b(c - a)}{ac} )So, ( d = frac{b(c - a)}{2ac} )Let me substitute this into the equation ( -b c d + a c d^2 = a^2 - c^2 )First, compute each term:( -b c d = -b c times frac{b(c - a)}{2ac} = -frac{b^2 c (c - a)}{2ac} = -frac{b^2 (c - a)}{2a} )( a c d^2 = a c times left( frac{b(c - a)}{2ac} right)^2 = a c times frac{b^2 (c - a)^2}{4a^2 c^2} = frac{b^2 (c - a)^2}{4a c} )So, substituting back:( -frac{b^2 (c - a)}{2a} + frac{b^2 (c - a)^2}{4a c} = a^2 - c^2 )Let me factor out ( frac{b^2 (c - a)}{4a c} ) from the left side:( frac{b^2 (c - a)}{4a c} left( -2c + (c - a) right) = a^2 - c^2 )Simplify inside the brackets:( -2c + c - a = -c - a = -(a + c) )So, left side becomes:( frac{b^2 (c - a)}{4a c} times -(a + c) = -frac{b^2 (c - a)(a + c)}{4a c} )Therefore, the equation is:( -frac{b^2 (c - a)(a + c)}{4a c} = a^2 - c^2 )Note that ( a^2 - c^2 = -(c^2 - a^2) = -(c - a)(c + a) ). So, right side is ( -(c - a)(c + a) ).So, we have:( -frac{b^2 (c - a)(a + c)}{4a c} = -(c - a)(c + a) )We can cancel ( -(c - a)(c + a) ) from both sides, assuming ( c neq a ) and ( c + a neq 0 ). Wait, but if ( c = a ), then ( c - a = 0 ), which would make the left side zero, but the right side would be ( -(0)(c + a) = 0 ), so it's possible. But let's see.If ( c = a ), then the original quadratics would be ( ax^2 + bx + a = 0 ) and ( ax^2 + bx + a = 0 ), which are the same equation. But the problem states that both quadratics have two distinct real solutions, so they must have different roots? Wait, no, they can have the same roots if ( a = c ). But in that case, the roots would be the same, but the problem says ( p_1, q_1, p_2, q_2 ) are in AP. If all roots are same, they can't form a proper AP with distinct terms. So, ( c neq a ).Similarly, if ( a + c = 0 ), then ( c = -a ). Let's see what happens.But let's proceed.After canceling ( -(c - a)(c + a) ) from both sides, we get:( frac{b^2}{4a c} = 1 )So,( frac{b^2}{4a c} = 1 )Which implies:( b^2 = 4a c )But wait, for a quadratic equation ( ax^2 + bx + c = 0 ), the discriminant is ( b^2 - 4ac ). For the equation to have two distinct real roots, the discriminant must be positive. So, ( b^2 - 4ac > 0 ). But if ( b^2 = 4ac ), then the discriminant would be zero, which contradicts the fact that the roots are distinct. Therefore, this leads to a contradiction unless our assumption is wrong.Wait, so if ( frac{b^2}{4a c} = 1 ), then ( b^2 = 4a c ), which would make the discriminant zero, but the problem states that both quadratics have two distinct real roots. Therefore, this is impossible.This suggests that our earlier step of canceling ( -(c - a)(c + a) ) might not be valid because if ( a + c = 0 ), then ( c + a = 0 ), making the term zero, and we can't cancel it. So, perhaps ( a + c = 0 ) is the solution.Let me check that.If ( a + c = 0 ), then ( c = -a ). Let's substitute ( c = -a ) into our earlier equations.First, equation (1):( 2p_1 + 2d = -frac{b}{a} )Equation (2):( 2p_1 + 4d = -frac{b}{c} = -frac{b}{-a} = frac{b}{a} )So, equation (2) becomes:( 2p_1 + 4d = frac{b}{a} )Subtract equation (1) from equation (2):( (2p_1 + 4d) - (2p_1 + 2d) = frac{b}{a} - (-frac{b}{a}) )Simplify:( 2d = frac{2b}{a} )So,( d = frac{b}{a} )Now, from equation (1):( 2p_1 + 2d = -frac{b}{a} )Substitute ( d = frac{b}{a} ):( 2p_1 + 2 times frac{b}{a} = -frac{b}{a} )So,( 2p_1 = -frac{b}{a} - frac{2b}{a} = -frac{3b}{a} )Thus,( p_1 = -frac{3b}{2a} )Now, ( p_2 = p_1 + 2d = -frac{3b}{2a} + 2 times frac{b}{a} = -frac{3b}{2a} + frac{2b}{a} = frac{b}{2a} )Similarly, ( q_1 = p_1 + d = -frac{3b}{2a} + frac{b}{a} = -frac{b}{2a} )And ( q_2 = p_2 + d = frac{b}{2a} + frac{b}{a} = frac{3b}{2a} )So, the four terms are:( p_1 = -frac{3b}{2a} ), ( q_1 = -frac{b}{2a} ), ( p_2 = frac{b}{2a} ), ( q_2 = frac{3b}{2a} )These form an arithmetic progression with common difference ( d = frac{b}{a} ), which is consistent.Now, let's check the products of the roots.For the first equation, ( p_1 p_2 = left( -frac{3b}{2a} right) left( frac{b}{2a} right) = -frac{3b^2}{4a^2} )But the product should be ( frac{c}{a} = frac{-a}{a} = -1 )So,( -frac{3b^2}{4a^2} = -1 )Multiply both sides by -1:( frac{3b^2}{4a^2} = 1 )So,( 3b^2 = 4a^2 )Similarly, for the second equation, ( q_1 q_2 = left( -frac{b}{2a} right) left( frac{3b}{2a} right) = -frac{3b^2}{4a^2} )But the product should be ( frac{a}{c} = frac{a}{-a} = -1 )So,( -frac{3b^2}{4a^2} = -1 )Which again gives ( 3b^2 = 4a^2 )So, both conditions lead to ( 3b^2 = 4a^2 ), which is consistent.Therefore, as long as ( a + c = 0 ), the conditions are satisfied, and the arithmetic progression holds.Hence, we've shown that ( a + c = 0 ).**Final Answer**boxed{a + c = 0}

answer：Okay, so I have this problem here about figuring out how many planks are needed to cover the floor of a square room, and also determining how many vershoks are in one arshin. Hmm, arshins and vershoks... I think those are old Russian units of measurement, right? I'm not too familiar with them, but I guess I can work with the information given.Alright, let's break down the problem. The room is square with a side length of 12 arshins. So, the area of the room would be 12 arshins multiplied by 12 arshins, which is 144 square arshins. That seems straightforward.Now, each plank is 6 arshins long and 6 vershoks wide. I need to figure out how many of these planks are needed to cover the entire floor. But wait, the problem also asks me to determine how many vershoks are in one arshin. So, it seems like I need to find the conversion factor between vershoks and arshins based on the information provided.Let me think about this. If I can find out how many vershoks are in one arshin, I can convert the width of the plank from vershoks to arshins, and then calculate the area of one plank in square arshins. Once I have the area of one plank, I can divide the total area of the room by the area of one plank to find out how many planks are needed.But the problem already tells me that 64 planks are needed. So, maybe I can use that information to find the conversion factor between vershoks and arshins. Let me set up an equation for that.First, let's denote the number of vershoks in one arshin as 'v'. So, 1 arshin = v vershoks. That means the width of the plank, which is 6 vershoks, is equal to 6/v arshins.Now, the area of one plank would be its length multiplied by its width. The length is 6 arshins, and the width is 6/v arshins. So, the area of one plank is 6 * (6/v) = 36/v square arshins.Since 64 planks are needed to cover the entire floor, the total area covered by the planks would be 64 * (36/v) square arshins. This total area should be equal to the area of the room, which is 144 square arshins.So, I can set up the equation: 64 * (36/v) = 144.Now, let's solve for 'v'. First, multiply both sides by 'v' to get rid of the denominator: 64 * 36 = 144v.Calculating 64 * 36: 64 * 30 is 1920, and 64 * 6 is 384, so 1920 + 384 = 2304. So, 2304 = 144v.Now, divide both sides by 144 to solve for 'v': v = 2304 / 144.Calculating that: 144 * 16 = 2304. So, v = 16.Therefore, there are 16 vershoks in one arshin.Wait, let me double-check that. If 1 arshin is 16 vershoks, then the width of each plank is 6 vershoks, which is 6/16 arshins, or 3/8 arshins. The area of one plank would then be 6 arshins * 3/8 arshins = 18/8 = 2.25 square arshins. If each plank is 2.25 square arshins, then 64 planks would cover 64 * 2.25 = 144 square arshins, which matches the area of the room. So, that checks out.So, it looks like my calculation is correct. There are 16 vershoks in one arshin.